Beyond symmetry groups: A grouping study on Escher's euclidean ornaments

Copyright © ACM.From art to science, ornaments constructed by repeating a base motif (tiling) have been a part of human culture. These ornaments exhibit various kinds of symmetries depending on the construction process as well as the symmetries of the base motif. The scientific study of the ornaments is the study of symmetry, i.e., the repetition structure. There is, however, an artistic side of the problem too: intriguing color permutations, clever choices of asymmetric interlocking forms, several symmetry breaking ideas, all that come with the artistic freedom. In this paper, in the context of Escher's Euclidean ornaments, we study ornaments without reference to fixed symmetry groups. We search for emergent categorical relations among a collection of tiles. We explore how these relations are affected when new tiles are inserted to the collection. We ask and answer whether it is possible to code symmetry group information implicitly without explicitly extracting the repetition structure, grids and motifs